We discuss a series of numerical experiments on the dispersion of neutrally buoyant particles in two-dimensional turbulent flows. The topology of two-dimensional turbulence is parametrized in terms of the relative dominance of deformation or rotation; this leads to a segmentation of the turbulent field into hyperbolic and elliptic domains. We show that some of the characteristic structural domains of two-dimensional turbulent flows, namely coherent structures and circulation cells, generate particle traps and peculiar accelerations which induce several complex properties of the particle dispersion processes at intermediate times. In general, passive particles are progressively pushed from the coherent structures and tend to concentrate in highly hyperbolic regions in the proximity of the isolines of zero vorticity. For large dispersion times, the background turbulent field is a privileged domain of particle richness; there is however a permanent particle exchange between the background field and the energetic circulation cells which surround the coherent structures. At intermediate times, an anomalous dispersion regime may appear, depending upon the relative weight of the different topological domains active in two-dimensional turbulence. The use of appropriate conditional averages allows the basic topology of two-dimensional turbulence to be characterized from a Lagrangian point of view. In particular, an intermediate $t^{\frac{5}{4}}$ anomalous dispersion law is shown to be associated with the action of hyperbolic regions where deformation dominates rotation; the motion of the advected particles in strongly elliptic regions where rotation dominates over deformation is shown to be associated with a $<e1>t<sup>t^{\frac{5}{3}}$ dispersion law. Because neutral particles concentrate on average in hyperbolic regions, the $<e1>t^{\frac{5}{4}}$ dispersion law is quite robust and it can be observed under very general circumstances.